#6459780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,20)(17,22)(18,28)(19,23)(21,27)(25,26);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,26,19,27,18,16)(14,20,28,21,23,25,22,24);(6,10)(7,11)(13,16,17,15,19,26,18,27)(14,21,28,25,23,24,22,20);(7,11)(9,12)(13,14)(15,25)(16,21)(17,28)(18,22)(19,23)(20,27)(24,26);(13,17,19,18)(14,22,23,28)(15,26,27,16)(20,24,25,21)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366976);